Graph Structure and Monadic Second-Order Logic: Language Theoretical Aspects

Graph structure is a flexible concept covering many different types of graph properties. Hierarchical decompositions yielding the notions of tree-width and clique-width, expressed by terms written with appropriate graph operations and associated with Monadic Second-order Logic are important tools for the construction of Fixed-Parameter Tractable algorithms and also for the extension of methods and results of Formal Language Theory to the description of sets of finite graphs. This informal overview presents the main definitions, results and open problems and tries to answer some frequently asked questions. Tree-width and monadic second-order (MS) logic are well-known tools for constructing fixed-parameter tractable (FPT) algorithms taking tree-width as parameter. Clique-width is, like tree-width, a complexity measure of graphs from which FPT algorithms can be built, in particular for problems specified in MS logic. These notions are thus essential for constructing (at least theoretically) tractable algorithms but also in the following three research fields: the study of the structure of graphs excluding induced subgraphs, minors or vertex-minors (a notion related to clique-width, see [48] or [18]); the extension of language theoretical notions in order to describe and to transform sets of finite and even countable graphs; the investigation of classes of finite and countable graphs on which MS logic is decidable. Although these four research fields have been initially developed independently, they are now more and more related. In particular, new structural results for graph classes have consequences for algorithmic applications (see [7]). This overview deals only with finite graphs, trees and relational structures. There is a rich theory of countable graphs described by logical formulas, logically defined transformations, equation systems and finite automata. The survey [1] is a good approach of this theory. Graph structure and logic Supported by the GRAAL project of "Agence Nationale pour la Recherche".